Consider the function f(x)=e^ kx (x^ 2 -4x+3)

MATHEMATICS · Multi-part · AP · Calculus AB · Analytical Applications of Differentiation · Analytical Applications of Differentiation

Problem

Consider the function

f(x)=e^{kx}(x^{2}-4x+3),

where

k\in\mathbb{R}

. (a) Find the first derivative of

f

and hence obtain the critical points in terms of

k

. (b) Show that

f

can have at most two critical points. (c) Determine all values of

k

for which

f

possesses exactly two critical points and each of them corresponds to a local minimum (that is, the curve has no local maximum).